Алгебра: Log4 (x-2)+log1/2 (x-2)=1/2

Решите задачу:

$log_4 (x-2)+log_\frac{1}{2} (x-2)=\frac{1}{2}\\ log_{2^2} (x-2)+log_{2^{-1}} (x-2)=\frac{1}{2}\\ 2log_{2} (x-2)-log_{2} (x-2)=\frac{1}{2}\\\\ log_{2} (x-2)^2-log_{2} (x-2)=\frac{1}{2}*log_2(2)\\ log_2 (\frac{(x-2)^2}{x-2} )=log_2( \sqrt{2} )\\\\ \left \{ {{\frac{(x-2)^2}{x-2}= \sqrt{2} } \atop {x-2\ \textgreater \ 0}} \right. \\ \left \{ {{x-2= \sqrt{2} } \atop {x\ \textgreater \ 2}} \right. \\ x=2+ \sqrt{2}$